Consider the matrices 0 7 0 8 0 0 0 0 1 4 13 -1 7 1 0 3 1 2 -1 25 -3 -19 1 -2 -1 A = 3 -6 1 1 1 and R = 1 4 -1 2 -7 1 -5 10 0 0 1 9 1 4 1 17 34 You may assume that R is the reduced echelon form (RREF) of A. Let ā1,. of A, in left-to-right order. ..., ā7 E R° denote the columns (a) Which of the vectors ā3, ā4, and āz belong to the span of ā1 and ā2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ā1,. No justification necessary. ,ā7? (c) Does āz belong to the span of ā1, đ2, and ā4? If not, explain why not; if so, write āz explicitly as a linear combination of ā1, ã2, and đ4. (d) Are ā1, ā2, ā6, and ā7 linearly independent? If so, explain why; if not, provide a nontrivial linear dependence relation among these four vectors (this means an equation of the form ciải + c2ã2 + Cóã6 + czā7 = 0 with at least one of the coefficients nonzero).
Consider the matrices 0 7 0 8 0 0 0 0 1 4 13 -1 7 1 0 3 1 2 -1 25 -3 -19 1 -2 -1 A = 3 -6 1 1 1 and R = 1 4 -1 2 -7 1 -5 10 0 0 1 9 1 4 1 17 34 You may assume that R is the reduced echelon form (RREF) of A. Let ā1,. of A, in left-to-right order. ..., ā7 E R° denote the columns (a) Which of the vectors ā3, ā4, and āz belong to the span of ā1 and ā2? No justification necessary. (b) What is the largest number of linearly independent vectors that can be chosen from among ā1,. No justification necessary. ,ā7? (c) Does āz belong to the span of ā1, đ2, and ā4? If not, explain why not; if so, write āz explicitly as a linear combination of ā1, ã2, and đ4. (d) Are ā1, ā2, ā6, and ā7 linearly independent? If so, explain why; if not, provide a nontrivial linear dependence relation among these four vectors (this means an equation of the form ciải + c2ã2 + Cóã6 + czā7 = 0 with at least one of the coefficients nonzero).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Please explain part a,c and d using the answer key I have attached below.
Transcribed Image Text:Consider the matrices
2
1
4
13
-1
7
|1 0
3
7
0 8
1
-1
25
-3
-19
1
-2 0
-1
0 0
0 0
0 0
A =
3
-6
1
1
1
9.
and
R =
1
4
-1
-7
1
-5
2
10
1
9.
1
4
1
17
2
34
0 0
ā7 E R° denote the columns
You may assume that R is the reduced echelon form (RREF) of A. Let ā1,
of A, in left-to-right order.
....
(a) Which of the vectors ā3, ā4, and ā, belong to the span of đ1 and ã2? No justification necessary.
(b) What is the largest number of linearly independent vectors that can be chosen from among ả1, .
No justification necessary.
,ā7?
(c) Does āz belong to the span of đ1, ã2, and ā4? If not, explain why not; if so, write āz explicitly as a
linear combination of ā1, ã2, and ā4.
(d) Are ā1, ā2, ã6, and ā7 linearly independent? If so, explain why; if not, provide a nontrivial linear
dependence relation among these four vectors (this means an equation of the form
ciãi + c2d2 + coã6 + c7ã7
with at least one of the coefficients nonzero).
Transcribed Image Text:For part (a), only đz belongs to span{ã1,ã2}; đ4 and āz do not. For part (b), the largest such number is 4,
since the matrix has four pivot columns. For part (c), the answer is yes, āz does belong to span{ā1, ã2, đ4}:
an explicit representation is āz = 7ã1 – ã2 + 4ã4. For part (d), the answer is no, ā1, d2, ā6, and ā7 are not
linearly independent. A nontrivial linear dependence relation among them is 8a1 + Oã2 + 9ã6 – ả7 = 0.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
